First, some books (maybe Hodge's, I don't have it right now) call the class of all finite structures embeddable in a structure, $\mathcal M,$ the Age of $\mathcal M.$ Sometimes this definition is changed to finitely generated rather than finite, like if say you wanted to talk about abelian groups. But, since your language is relational, that does not matter. Any Age satisfies some model theoretic properties, which are likely in Hodge's book, Hereditary and Joint Embedding (in fact Fraisse proved that these were necessary and sufficient conditions for being an Age). If the class you are considering is an Age and also has the amalgamation property, then it is true that there is a unique countable homogeneous model with that Age.
So, we should get back to your question. It is clear that the class of finite linear orderings has the amalgamation property. By the previous paragraph, you need only show that the rationals are a countable structure with the Age being the class of all finite linear orderings and that the rationals are homogeneous. So, the only potentially difficult thing is to prove homogeneity. To do this, you need only show that given a finite partial map $f: \mathbb Q \rightarrow \mathbb Q$ which respects the order can be extended to an isomorphism.